大交集的Frostman型引理及其在丢番图逼近中的应用
A Frostman type lemma for sets with large intersections, and an application to Diophantine approximation
摘要：
我们认为类$\mathscr{G}^s ([0,1])$的子集$[0,1]$，最初介绍了法尔科纳，是封闭的可数交叉，这样，在类中的每一个集合的Hausdorff维数至少$s$。我们提供了一个Frostman型引理，以确定是否limsup-set是在这样的类。假设$E = \limsup E_n \subset [0,1]$和$μ_n$是在$E_n$中支持的概率测度。如果有一个常数$C$，使\[\iint|x-y|^{-s}\, \mathrm{d}μ_n(x)\mathrm{d}μ_n(y)<C\]的所有$n$，然后在适当的条件下的极限测度序列$(μ_n)$，我们证明了集合$E$是在类$\mathscr{G}^s ([0,1])$。作为一个应用，我们证明了，对于$α> 1$和几乎所有的$λ\in (\frac{1}{2},1)$的集合\[ E_λ(α) = \{\,x\in[0,1] : |x - s_n| < 2^{-αn} \text{infinitely often}\ \}\]，其中$s_n \in \{\,(1-λ)\sum_{k=0}^na_kλ^k$和$a_k\in\{0,1\}\,\}$，属于类$\mathscr{G}^s$为$s \leq \frac{1}α$。这改进了我们以前的一个结果。
英文摘要：
We consider classes $\mathscr{G}^s ([0,1])$ of subsets of $[0,1]$, originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least $s$. We provide a Frostman type lemma to determine if a limsup-set is in such a class. Suppose $E = \limsup E_n \subset [0,1]$, and that $μ_n$ are probability measures with support in $E_n$. If there is a constant $C$ such that \[\iint|x-y|^{-s}\, \mathrm{d}μ_n(x)\mathrm{d}μ_n(y)<C\] for all $n$, then under suitable conditions on the limit measure of the sequence $(μ_n)$, we prove that the set $E$ is in the class $\mathscr{G}^s ([0,1])$. As an application we prove that for $α> 1$ and almost all $λ\in (\frac{1}{2},1)$ the set \[ E_λ(α) = \{\,x\in[0,1] : |x - s_n| < 2^{-αn} \text{infinitely often}\ \}\] where $s_n \in \{\,(1-λ)\sum_{k=0}^na_kλ^k$ and $a_k\in\{0,1\}\,\}$, belongs to the class $\mathscr{G}^s$ for $s \leq \frac{1}α$. This improves one of our previous results.
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